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In [1] a verification of the discrete Fourier transform pairs was performed. A much different looking discrete Fourier transform pair is given in [2] \$ A.4. This transform pair samples the points at what are called the Nykvist time instants given by

\label{eqn:discreteFourier:20}
t_k = \frac{T k}{2 N + 1}, \qquad k \in [-N, \cdots N]

Note that the endpoints of these sampling points are not $$\pm T/2$$, but are instead at

\label{eqn:discreteFourier:40}
\pm \frac{T}{2} \inv{1 + 1/N},

which are slightly within the interior of the $$[-T/2, T/2]$$ range of interest. The reason for this slightly odd seeming selection of sampling times becomes clear if one calculate the inversion relations.

Given a periodic ($$\omega_0 T = 2 \pi$$) bandwith limited signal evaluated only at the Nykvist times $$t_k$$,

\label{eqn:discreteFourier:60}
\boxed{
x(t_k) = \sum_{n = -N}^N X_n e^{ j n \omega_0 t_k},
}

assume that an inversion relation can be found. To find $$X_n$$ evaluate the sum

\label{eqn:discreteFourier:80}
\begin{aligned}
&\sum_{k = -N}^N x(t_k) e^{-j m \omega_0 t_k} \\
\sum_{k = -N}^N
\lr{
\sum_{n = -N}^N X_n e^{ j n \omega_0 t_k}
}
e^{-j m \omega_0 t_k} \\
\sum_{n = -N}^N X_n
\sum_{k = -N}^N
e^{ j (n -m )\omega_0 t_k}
\end{aligned}

This interior sum has the value $$2 N + 1$$ when $$n = m$$. For $$n \ne m$$, and
$$a = e^{j (n -m ) \frac{2 \pi}{2 N + 1}}$$, this is

\label{eqn:discreteFourier:100}
\begin{aligned}
\sum_{k = -N}^N
e^{ j (n -m )\omega_0 t_k}
&=
\sum_{k = -N}^N
e^{ j (n -m )\omega_0 \frac{T k}{2 N + 1}} \\
&=
\sum_{k = -N}^N a^k \\
&=
a^{-N} \sum_{k = -N}^N a^{k+ N} \\
&=
a^{-N} \sum_{r = 0}^{2 N} a^{r} \\
&=
a^{-N} \frac{a^{2 N + 1} – 1}{a – 1}.
\end{aligned}

Since $$a^{2 N + 1} = e^{2 \pi j (n – m)} = 1$$, this sum is zero when $$n \ne m$$. This means that

\label{eqn:discreteFourier:120}
\sum_{k = -N}^N
e^{ j (n -m )\omega_0 t_k} = (2 N + 1) \delta_{n,m},

which provides the desired Fourier inversion relation

\label{eqn:discreteFourier:140}
\boxed{
X_m = \inv{2 N + 1} \sum_{k = -N}^N x(t_k) e^{-j m \omega_0 t_k}.
}

# References

[1] Peeter Joot. Condensed matter physics., appendix: Discrete Fourier transform. 2013. URL https://peeterjoot.com/archives/math2013/phy487.pdf. [Online; accessed 02-December-2014].

[2] Giannini and Leuzzi Nonlinear Microwave Circuit Design. Wiley Online Library, 2004.